On the bit-error probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization

We investigate the differences between allowable differential encoding strategies and their associated bit-error probability performances for quadrature phase-shift keying (QPSK) and offset QPSK modulations when the carrier demodulation reference signals are supplied by the optimum (motivated by maximum a posteriori estimation of carrier phase) carrier-tracking loop suitable for that modulation. In particular, we show that in the presence of carrier-synchronization phase ambiguity but an otherwise ideal loop, both the symbol and bit-error probabilities in the presence of differential encoding are identical for the two modulations. On the other hand, when in addition the phase error introduced by the loop´s finite signal-to-noise ratio is taken into account, it is shown that the two differentially encoded modulations behave differently, and their performances are no longer equivalent. A similar statement has previously been demonstrated for the same modulations when the phase ambiguity was assumed to have been perfectly resolved by means other than differential encoding.